Find all the real solutions to cubic equation, Mathematics

Use the fixed point iteration to find the fixed point(s) for the function g(x) = 1 + x - x^2/3

Find all the real solutions to cubic equation x^3 +4x^2-10=0. Use the cubic equation x^3 + 4x^2 - 10 =0 and perform the following call to the regulaFalsi [0, 1, 30]

Use newton's method to find the three roots of a cubic polynomial f(x) = 4x^3 - 15x^2 + 17x-6. Determine the Newton-raphson iteration formula g[x] = x - (f(x)/f'(x)) that is used. Show details of the computation for the starting value p0 = 3.

Use the secant method to find the three roots of cubic polynomial f[x]=4x^3 - 16x^2 + 17x - 4. Determine the secant iterative formula g[x] = x - (f[x]/f'[x]) that is used. Show details of the computation for the starting value p0=3 and p1=2.8

Use appropiate Lagrange interpolating polynomials of degrees one, two, and three to approximate each of the following:

F'(x)= (f(x+h)-f(x-h))/2h

F'(x)= (-f(x+2h)+8f(x+h)-8f(x-h)+f(x-2h))/12h

Consider the richardson table for derivatives in the form step size table

Step size table

H D(0,0)

H/2 D(1,0) D(1,1)

H/2^2 D(2,0) D(2,1) D(2,2)

H/2^3 D(3,0) D(3,1) D(2,3) D(3,3)

.

.

Where the central difference formula

(h) = (f(x+h)-f(x-h)) /2h

Is used to construct the first column using

D(n,0)= (h/2^n)

And the following formula

D(n,m)= (4^mD(n,m-1)-D(n-1,m-1))/4^m-1 (use for hand calculations)

D(n,m-1)+((D(n,m-1)-D(n-1,m-1)/(4^m-1)) (use for programming)

Is used, for n≥m, to obtain entries in other columns in terms of the entry to their left and the entry above this entry. For example, D(2,1) is obtained in terms of D(2,0) and D(1,0) and D(3,2) is obtained in terms of D(3,1) and D(2,1)

A) construct the table for the derivative of tan x at x=0.5. Choose an initial step size of h=1 and calculate 4 rows by hand using a calculator